# idnlarx

Nonlinear ARX model

## Description

An `idnlarx`

model represents a nonlinear ARX model, which is an
extension of the linear ARX structure and contains linear and nonlinear
functions.

A nonlinear ARX model consists of model regressors and an output function. The output function
contains one or more *mapping objects*, one for each model output. Each
mapping object can include a linear and a nonlinear function that act on the model
regressors to give the model output and a fixed offset for that output. This block diagram
represents the structure of a single-output nonlinear ARX model in a simulation
scenario.

The software computes the nonlinear ARX model output *y* in
two stages:

It computes regressor values from the current and past input values and the past output data.

In the simplest case, regressors are delayed inputs and outputs, such as

*u*(*t*–1) and*y*(*t*–3). These kind of regressors are called*linear regressors*. You specify linear regressors using the`linearRegressor`

object. You can also specify linear regressors by using linear ARX model orders as an input argument. For more information, see Nonlinear ARX Model Orders and Delay. However, this second approach constrains your regressor set to linear regressors with consecutive delays. To create*polynomial regressors*, use the`polynomialRegressor`

object. To create*periodic regressors*that contain the sine and cosine functions of delayed input and output variables , use the`periodicRegressor`

object. You can also specify*custom regressors*, which are nonlinear functions of delayed inputs and outputs. For example,*u*(*t*–1)*y*(*t*–3) is a custom regressor that multiplies instances of input and output together. Specify custom regressors using the`customRegressor`

object.You can assign any of the regressors as inputs to the linear function block of the output function, the nonlinear function block, or both.

It maps the regressors to the model output using an output function block. The output function block can include multiple mapping objects, with each mapping object containing linear, nonlinear, and offset blocks in parallel. For example, consider the following equation:

$$F(x)={L}^{T}(x-r)+g\left(Q(x-r)\right)+d$$

Here,

*x*is a vector of the regressors, and*r*is the mean of*x*. $$F(x)={L}^{T}(x-r)+{y}_{0}$$ is the output of the linear function block. $$g\left(Q(x-r)\right)+{y}_{0}$$ represents the output of the nonlinear function block.*Q*is a projection matrix that makes the calculations well-conditioned.*d*is a scalar offset that is added to the combined outputs of the linear and nonlinear blocks. The exact form of*F*(*x*) depends on your choice of output function. You can select from the available mapping objects, such as tree-partition networks, wavelet networks, and multilayer neural networks. You can also exclude either the linear or the nonlinear function block from the output function.When estimating a nonlinear ARX model, the software computes the model parameter values, such as

*L*,*r*,*d*,*Q*, and other parameters specifying*g*.

The resulting nonlinear ARX models are `idnlarx`

objects that store all model data, including model regressors and
parameters of the output function. For more information about these objects, see Nonlinear Model Structures.

For more information on the `idnlarx`

model structure, see What are Nonlinear ARX Models?.

For `idnlarx`

object properties, see Properties.

## Creation

You can obtain an `idnlarx`

object in one of two ways.

### Syntax

### Description

#### Specify Model Directly

specifies a set of linear regressors using ARX model orders. Use this syntax when you
extend an ARX linear model, or when you plan to use only regressors that are linear with
consecutive lags.`sys`

= idnlarx(`output_name`

,`input_name`

,`orders`

)

creates a nonlinear ARX model with the output and input names of
`sys`

= idnlarx(`output_name`

,`input_name`

,Regressors)`output_name`

and `input_name`

, respectively,
and a regressor set in `Regressors`

that contains any combination of linear, polynomial, periodic,
and custom regressors. The software constructs `sys`

using the
default wavelet network (`'idWaveletNetwork'`

) mapping object for the
output function.

#### Initialize Model Values Using Linear Model

uses a linear model `sys`

= idnlarx(`linmodel`

)`linmodel`

to extract certain properties such as
names, units, and sample time, and to initialize the values of the linear coefficients
of the model. Use this syntax when you want to create a nonlinear ARX model as an
extension of, or an improvement upon, an existing linear model.

#### Specify Model Properties

specifies additional properties of
the `sys`

= idnlarx(___,`Name,Value`

)`idnlarx`

model structure using one or more name-value arguments.

### Input Arguments

`orders`

— ARX model orders

`nlarx`

orders `[na nb nk]`

ARX model orders, specified as the matrix `[na nb nk]`

.
`na`

denotes the number of delayed outputs, `nb`

denotes the number of delayed inputs, and `nk`

denotes the minimum
input delay. The minimum output delay is fixed to `1`

. For more
information on how to construct the `orders`

matrix, see `arx`

.

When you specify `orders`

, the software converts the order
information into a linear regressor form in the `idnlarx`

`Regressors`

property. For an example, see Create Nonlinear ARX Model Using ARX Model Orders.

`linmodel`

— Discrete-time linear model

`idpoly`

object | `idss`

object | `idtf`

object | `idproc`

object

Discrete-time identified input/output linear model, specified as any linear model
created using estimators, that is, an `idpoly`

object, an `idss`

object, an `idtf`

object, or an `idproc`

object with `Ts`

> 0. Create this model using the
constructor function for the object or estimate the model using the associated
estimation command. For example, to create an ARX model, use `arx`

, and specify the resulting `idpoly`

object as
`linmodel`

.

## Properties

`Regressors`

— Regressor specification

`linearRegressor`

object | `polynomialRegressor`

object | `periodicRegressor`

object | `customRegressor`

object | column array of regressor specification objects

Regressor specification, specified as a column vector containing one or more
regressor specification objects, which are the `linearRegressor`

objects, `polynomialRegressor`

objects, `periodicRegressor`

objects, and `customRegressor`

objects. Each object specifies a formula for generating
regressors from lagged variables. For example:

`L = linearRegressor({'y1','u1'},{1,[2 5]})`

generates the regressors*y*(_{1}*t*–1),*u*(_{1}*t*–2), and*u*(_{2}*t*–5).`P = polynomialRegressor('y2',4:7,2)`

generates the regressors*y*(_{2}*t*–4)^{2},*y*(_{2}*t*–5)^{2},*y*(_{2}*t*–6)^{2}, and*y*(_{2}*t*–7)^{2}.`SC = periodicRegressor({'y1','u1'},{1,2})`

generates the regressors*y*(_{1}*t*-1)), cos(*y*(_{1}*t*-1)), sin(*u*(_{1}*t*-2)), and cos(*u*(_{1}*t*-2)).`C = customRegressor({'y1','u1','u2'},{1 2 2},@(x,y,z)sin(x.*y+z))`

generates the single regressor sin(*y*(_{1}*t*–1)*u*(_{1}*t*–2)+*u*(_{2}*t*–2).

For an example that implements these regressors, see Create and Combine Regressor Types.

To add regressors to an existing model, create a vector of specification objects and
use dot notation to set `Regressors`

to this vector. For example, the
following code first creates the `idnlarx`

model `sys `

and then adds the regressor objects `L`

, `P`

,
`SC`

, and `C`

to the regressors of
`sys`

.

sys = idnlarx({'y1','y2'},{'u1','u2'}); R = [L;P;SC;C]; sys.Regressors = R;

For an example of creating and using a linear regressor set, see Create Nonlinear ARX Model Using Linear Regressors.

`OutputFcn`

— Output function

`'idWaveletNetwork'`

(default) | `'idLinear'`

| `[]`

| `''`

| `'idSigmoidNetwork'`

| `'idTreePartition'`

| `'idGaussianProcess'`

| `'idTreeEnsemble'`

| `'idSupportVectorMachine'`

| mapping object | array of mapping objects

Output function that maps the regressors of the `idnlarx`

model into
the model output, specified as a column array containing zero or more of the following
strings or mapping objects:

`'idWaveletNetwork'` or `idWaveletNetwork` object | Wavelet network |

`'idLinear'` or `''` or
`[]` or `idLinear` object | Linear function |

`'idSigmoidNetwork'` or `idSigmoidNetwork` object | Sigmoid network |

`'idTreePartition'` or `idTreePartition` object | Binary tree partition regression model |

`'idGaussianProcess'` or `idGaussianProcess` object | Gaussian process regression model (requires Statistics and Machine Learning Toolbox™) |

`'idTreeEnsemble'` or `idTreeEnsemble` | Regression tree ensemble model (requires Statistics and Machine Learning Toolbox) |

`'idSupportVectorMachine'` or `idSupportVectorMachine` | Kernel-based Support Vector Machine (SVM) regression model with constraints (requires Statistics and Machine Learning Toolbox) |

`'idNeuralNetwork'` or `idNeuralNetwork` object | Multilayer neural network (requires Statistics and Machine Learning Toolbox or Deep Learning Toolbox™) |

`idCustomNetwork` object | Custom network — Similar to `idSigmoidNetwork` , but
with a user-defined replacement for the sigmoid function |

The `idWaveletNetwork`

, `idSigmoidNetwork`

,
`idTreePartition`

, and `idCustomNetwork`

objects
contain both linear and nonlinear components. You can remove (not use) the linear
components of `idWaveletNetwork`

,
`idSigmoidNetwork`

, and `idCustomNetwork`

by
setting the `LinearFcn.Use`

value to
`false`

.

The `idTreeEnsemble`

and
`idSupportVectorMachine`

objects contain only a nonlinear
component. The `idLinear`

function, as the name implies, has only a
linear component.

Specifying a character vector, for example `'idSigmoidNetwork'`

,
creates a mapping object with default settings. Alternatively, you can specify mapping
object properties in two other ways:

Create the mapping object using arguments to modify default properties.

MO = idSigmoidNetwork(15)

Create a default mapping object first and then use dot notation to modify properties.

MO = idSigmoidNetwork; MO.NumberOfUnits = 15

For *n _{y}* output channels, you can specify
mapping objects individually for each channel by setting

`OutputFcn`

to an array of *n*mapping objects. For example, the following code specifies

_{y}`OutputFcn`

using dot notation
for a system with two input channels and two output channels.
sys = idnlarx({'y1','y2'},{'u1','u2'}); sys.OutputFcn = [idWaveletNetwork; idSigmoidNetwork]

`OutputFcn`

as a
character vector or a single mapping object.
`OutputFcn`

represents a static mapping function that transforms
the regressors of the nonlinear ARX model into the model output.
`OutputFcn`

is static because it does not depend on the time. For
example, if $$y(t)={y}_{0}+{a}_{1}y(t-1)+{a}_{2}y(t-2)+\dots +{b}_{1}u(t-1)+{b}_{2}u(t-2)+\dots $$, then `OutputFcn`

is a linear function represented
by the `idLinear`

object.

For an example of specifying the output function, see Specify Output Function for Nonlinear ARX Model.

`RegressorUsage`

— Regressor assignments

table with logical entries

Regressor assignments to the linear and nonlinear components of the nonlinear ARX
model, specified as an
*n _{r}*-by-

*n*table with logical entries that specify which regressors to use for which component. Here,

_{c}*n*is the number of regressors.

_{r}*n*is the total number of linear and nonlinear components in

_{c}`OutputFcn`

. The rows of the table correspond to individual regressors. The
row names are set to regressor names. If the table value for row *i*and component index

*j*is

`true`

, then the
*i*th regressor is an input to the linear or nonlinear component

*j*.

For multi-output systems, `OutputFcn`

contains one mapping object
for each output. Each mapping object can use both linear and nonlinear components or
only one of the two components.

For an example of viewing and modifying the `RegressorUsage`

property, see Modify Regressor Assignments to Output Function Components.

`Normalization`

— Regressor and output data centering and scaling

structure (default)

Regressor and output centering and scaling, specified as a structure. As the
following table shows, each field in the structure contains a row vector with a length
that is equal to the number of either regressors
(*n _{r}*) or model outputs
(

*n*).

_{y}Field | Description | Default Element Value |
---|---|---|

`RegressorCenter` | Row vector of length n_{r} | `NaN` |

`RegressorScale` | Row vector of length n_{r} | `NaN` |

`OutputCenter` | Row vector of length n_{y} | `NaN` |

`OutputScale` | Row vector of length n_{y} | `NaN` |

For a matrix `X`

, with centering vector `C`

and
scaling vector `S`

, the software computes the normalized form of
`X`

using `Xnorm = (X-C)./S`

.

The following figure illustrates the normalization flow for a nonlinear ARX model.

In this figure:

The algorithm converts the inputs

*u*(*t*) and*y*(*t*) into the regressor set*R*(*t*).The algorithm uses the regressor centering and scaling parameters to normalize

*R*(*t*) as*R*(_{N}*t*).*R*(_{N}*t*) provides the input to the mapping function, which then produces the normalized output*y*_{N}The algorithm uses the output scaling and centering parameters to restore the original range, producing

*y*(*t*).

Typically, the software normalizes the data automatically during model estimation,
in accordance with the option settings in `nlarxOptions`

for `Normalize`

and
`NormalizationOptions`

. You can also directly assign centering and
scaling values by specifying the values in vectors, as described in the previous table.
The values that you assign must be real and finite. This approach can be useful, for
example, when you are simulating your model using inputs that represent a different
operating point from the operating point for the original estimation data. You can
assign the values for any field independently. The software will estimate the values of
any fields that remain unassigned (`NaN`

).

`Report`

— Summary report

report field values

This property is read-only.

Summary report that contains information about the estimation options and results
for a nonlinear ARX model obtained using the `nlarx`

command. Use `Report`

to find estimation
information for the identified model, including:

Estimation method

Estimation options

Search termination conditions

Estimation data fit

The contents of `Report`

are irrelevant if the model was
constructed using `idnlarx`

.

sys = idnlarx('y1','u1',reg); sys.Report.OptionsUsed

ans = []

If you use `nlarx`

to estimate the model, the fields of
`Report`

contain information on the estimation data, options, and
results.

```
load iddata1;
sys = nlarx(z1,reg);
m.Report.OptionsUsed
```

Option set for the nlarx command: IterativeWavenet: 'auto' Focus: 'prediction' Display: 'off' Regularization: [1x1 struct] SearchMethod: 'auto' SearchOptions: [1x1 idoptions.search.identsolver] OutputWeight: 'noise' Advanced: [1x1 struct]

For more information on this property and how to use it, see Output Arguments in the `nlarx`

reference page and Estimation Report.

`TimeVariable`

— Independent variable

`'t'`

(default) | character vector

Independent variable for the inputs, outputs, and—when available—internal states, specified as a character vector.

`NoiseVariance`

— Noise variance

matrix

Noise variance (covariance matrix) of the model innovations *e*.
The estimation algorithm typically sets this property. However, you can also assign the
covariance values by specifying an `ny`

-by-`ny`

matrix.

`Ts`

— Sample time

`1`

(default) | positive scalar

Sample time, specified as a positive scalar representing the sampling period. This
value is expressed in the unit specified by the `TimeUnit`

property of
the model.

`TimeUnit`

— Units for time variable

`'seconds'`

(default) | `'nanoseconds'`

| `'microseconds'`

| `'milliseconds'`

| `'minutes'`

| `'hours'`

| `'days'`

| `'weeks'`

| `'months'`

| `'years'`

Units for the time variable, the sample time `Ts`

, and any time
delays in the model, specified as one of the following values:

`'nanoseconds'`

`'microseconds'`

`'milliseconds'`

`'seconds'`

`'minutes'`

`'hours'`

`'days'`

`'weeks'`

`'months'`

`'years'`

Changing this property has no effect on other properties, and therefore changes the
overall system behavior. Use `chgTimeUnit`

(Control System Toolbox) to convert between time units
without modifying system behavior.

`InputName`

— Input channel names

`''`

for all input channels (default) | character vector | cell array of character vectors

Input channel names, specified as one of the following:

Character vector — For single-input models, for example,

`'controls'`

.Cell array of character vectors — For multi-input models.

Input names in Nonlinear ARX models must be valid MATLAB^{®} variable names after you remove any spaces.

Alternatively, use automatic vector expansion to assign input names for multi-input
models. For example, if `sys`

is a two-input model, enter:

sys.InputName = 'controls';

The input names automatically expand to
`{'controls(1)';'controls(2)'}`

.

When you estimate a model using an `iddata`

object, `data`

, the software automatically sets
`InputName`

to `data.InputName`

.

You can use the shorthand notation `u`

to refer to the
`InputName`

property. For example, `sys.u`

is
equivalent to `sys.InputName`

.

Input channel names have several uses, including:

Identifying channels on model display and plots

Extracting subsystems of MIMO systems

Specifying connection points when interconnecting models

`InputUnit`

— Input channel units

`''`

for all input channels (default) | character vector | cell array of character vectors

Input channel units, specified as one of the following:

Character vector — For single-input models, for example,

`'seconds'`

.Cell array of character vectors — For multi-input models.

Use `InputUnit`

to keep track of input signal units.
`InputUnit`

has no effect on system behavior.

`InputGroup`

— Input channel groups

structure with no fields (default) | structure

Input channel groups. The `InputGroup`

property lets you assign the
input channels of MIMO systems into groups and refer to each group by name. Specify
input groups as a structure. In this structure, field names are the group names, and
field values are the input channels belonging to each group. For example:

sys.InputGroup.controls = [1 2]; sys.InputGroup.noise = [3 5];

creates input groups named `controls`

and `noise`

that include input channels 1, 2 and 3, 5, respectively. You can then extract the
subsystem from the `controls`

inputs to all outputs using:

sys(:,'controls')

`OutputName`

— Output channel names

`''`

for all output channels (default) | character vector | cell array of character vectors

Output channel names, specified as one of the following:

Character vector — For single-output models. For example,

`'measurements'`

.Cell array of character vectors — For multi-output models.

Output names in Nonlinear ARX models must be valid MATLAB variable names after you remove any spaces.

Alternatively, use automatic vector expansion to assign output names for
multi-output models. For example, if `sys`

is a two-output model,
enter:

sys.OutputName = 'measurements';

The output names automatically expand to
`{'measurements(1)';'measurements(2)'}`

.

When you estimate a model using an `iddata`

object, `data`

, the software automatically sets
`OutputName`

to `data.OutputName`

.

You can use the shorthand notation `y`

to refer to the
`OutputName`

property. For example, `sys.y`

is
equivalent to `sys.OutputName`

.

Output channel names have several uses, including:

Identifying channels on model display and plots

Extracting subsystems of MIMO systems

Specifying connection points when interconnecting models

`OutputUnit`

— Output channel units

`''`

for all output channels (default) | character vector | cell array of character vectors

Output channel units, specified as one of the following:

Character vector — For single-output models. For example,

`'seconds'`

.Cell array of character vectors — For multi-output models.

Use `OutputUnit`

to keep track of output signal units.
`OutputUnit`

has no effect on system behavior.

`OutputGroup`

— Output channel groups

structure with no fields (default) | structure

Output channel groups. The `OutputGroup`

property lets you assign
the output channels of MIMO systems into groups and refer to each group by name. Specify
output groups as a structure. In this structure, field names are the group names, and
field values are the output channels belonging to each group. For example:

sys.OutputGroup.temperature = [1]; sys.OutputGroup.measurement = [3 5];

creates output groups named `temperature`

and
`measurement`

that include output channels 1, and 3, 5, respectively.
You can then extract the subsystem from all inputs to the `measurement`

outputs using:

sys('measurement',:)

`Name`

— System Name

`''`

(default) | character vector

System name, specified as a character vector. For example, ```
'system
1'
```

.

`Notes`

— Notes on system

`0`

-by-`1`

string (default) | string | character vector

Any text that you want to associate with the system, specified as a string or a cell
array of character vectors. The property stores whichever data type you provide. For
instance, if `sys1`

and `sys2`

are dynamic system
models, you can set their `Notes`

properties as follows.

sys1.Notes = "sys1 has a string."; sys2.Notes = 'sys2 has a character vector.'; sys1.Notes sys2.Notes

ans = "sys1 has a string." ans = 'sys2 has a character vector.'

`UserData`

— Data to associate with system

`[]`

(default) | any MATLAB data type

Any data you want to associate with the system, specified as any MATLAB data type.

## Object Functions

For information about object functions for `idnlarx`

, see Nonlinear ARX Models.

## Examples

### Create Nonlinear ARX Model Using ARX Model Orders

Create an `idnlarx`

model by specifying an ARX model order vector.

Create an order vector of the form `[na nb nk]`

, where `na`

and `nb`

are the orders of the *A* and *B* ARX model polynomials and `nk`

is the number of input/output delays.

na = 2; nb = 3; nk = 5; orders = [na nb nk];

Construct a nonlinear ARX model `sys`

.

output_name = 'y1'; input_name = 'u1'; sys = idnlarx(output_name,input_name,[2 3 5]);

View the output function.

disp(sys.OutputFcn)

Wavelet Network Nonlinear Function: Wavelet network with number of units chosen automatically Linear Function: uninitialized Output Offset: uninitialized NonlinearFcn: '<Wavelet and scaling function units and their parameters>' LinearFcn: '<Linear function parameters>' Offset: '<Offset parameters>' EstimationOptions: '<Estimation options>'

By default, the model uses a wavelet network, represented by a `idWaveletNetwork`

object, for the output function. The `idWaveletNetwork`

object includes linear and nonlinear components.

View the `Regressors`

property.

disp(sys.Regressors)

Linear regressors in variables y1, u1 Variables: {'y1' 'u1'} Lags: {[1 2] [5 6 7]} UseAbsolute: [0 0] TimeVariable: 't'

The `idnlarx`

constructor transforms the model orders into the `Regressors`

form.

The L

`ags`

array for`y1`

,`[1 2]`

, is equivalent to the`na`

value of 2. Both forms specify two consecutive output regressors,`y1(t-1)`

and`y1(t-2)`

.The

`Lags`

array for`u1`

,`[5 6 7]`

, incorporates the three delays specified by the`nb`

value of 3, and shifts them by the`nk`

value of 5. The input regressors are therefore`u1(t-5)`

,`u1(t-6)`

, and`u1(t-7)`

.

View the regressors.

getreg(sys)

`ans = `*5x1 cell*
{'y1(t-1)'}
{'y1(t-2)'}
{'u1(t-5)'}
{'u1(t-6)'}
{'u1(t-7)'}

You can use the `orders`

syntax to specify simple linear regressors. However, to create more complex regressors, use the regressor commands `linearRegressor`

, `polynomialRegressor`

, and `customRegressor`

to create a combined regressor for the `Regressors`

syntax`.`

### Create Nonlinear ARX Model Using Linear Regressors

Construct an `idnlarx`

model by specifying linear regressors.

Create a linear regressor that contains two output lags and two input lags.

output_name = 'y1'; input_name = 'u1'; var_names = {output_name,input_name}; output_lag = [1 2]; input_lag = [1 2]; lags = {output_lag,input_lag}; reg = linearRegressor(var_names,lags)

reg = Linear regressors in variables y1, u1 Variables: {'y1' 'u1'} Lags: {[1 2] [1 2]} UseAbsolute: [0 0] TimeVariable: 't'

The model contains the regressors `y(t-1)`

, `y(t-2)`

, `u(t-1)`

, and `u(t-2)`

.

Construct the `idnlarx`

model and view the regressors.

sys = idnlarx(output_name,input_name,reg); getreg(sys)

`ans = `*4x1 cell*
{'y1(t-1)'}
{'y1(t-2)'}
{'u1(t-1)'}
{'u1(t-2)'}

View the output function.

disp(sys.OutputFcn)

Wavelet Network Nonlinear Function: Wavelet network with number of units chosen automatically Linear Function: uninitialized Output Offset: uninitialized NonlinearFcn: '<Wavelet and scaling function units and their parameters>' LinearFcn: '<Linear function parameters>' Offset: '<Offset parameters>' EstimationOptions: '<Estimation options>'

View the regressor usage table.

disp(sys.RegressorUsage)

y1:LinearFcn y1:NonlinearFcn ____________ _______________ y1(t-1) true true y1(t-2) true true u1(t-1) true true u1(t-2) true true

All the regressors are inputs to both the linear and nonlinear components of the `idWaveletNetwork`

object.

### Create and Configure Nonlinear ARX Model

Create a nonlinear ARX model with a linear regressor set.

Create a linear regressor that contains three output lags and two input lags.

output_name = 'y1'; input_name = 'u1'; var_names = {output_name,input_name}; output_lag = [1 2 3]; input_lag = [1 2]; lags = {output_lag,input_lag}; reg = linearRegressor(var_names,lags)

reg = Linear regressors in variables y1, u1 Variables: {'y1' 'u1'} Lags: {[1 2 3] [1 2]} UseAbsolute: [0 0] TimeVariable: 't'

Construct the nonlinear ARX model.

sys = idnlarx(output_name,input_name,reg);

View the `Regressors`

property.

disp(sys.Regressors)

Linear regressors in variables y1, u1 Variables: {'y1' 'u1'} Lags: {[1 2 3] [1 2]} UseAbsolute: [0 0] TimeVariable: 't'

`sys`

uses `idWavenetNetwork`

as the default output function. Reconfigure the output function to `idSigmoidNetwork`

.

```
sys.OutputFcn = 'idSigmoidNetwork';
disp(sys.OutputFcn)
```

Sigmoid Network Nonlinear Function: Sigmoid network with 10 units Linear Function: uninitialized Output Offset: uninitialized NonlinearFcn: '<Sigmoid units and their parameters>' LinearFcn: '<Linear function parameters>' Offset: '<Offset parameters>'

### Specify Output Function for Nonlinear ARX Model

Specify the sigmoid network output function when you construct a nonlinear ARX model.

Assign variable names and specify a regressor set.

output_name = 'y1'; input_name = 'u1'; r = linearRegressor({output_name,input_name},{1 1});

Construct a nonlinear ARX model that specifies the `idSigmoidNetwork`

output function. Set the number of terms in the sigmoid expansion to `15`

.

sys = idnlarx(output_name,input_name,r,idSigmoidNetwork(15));

View the output function specification.

disp(sys.OutputFcn)

Sigmoid Network Nonlinear Function: Sigmoid network with 15 units Linear Function: uninitialized Output Offset: uninitialized NonlinearFcn: '<Sigmoid units and their parameters>' LinearFcn: '<Linear function parameters>' Offset: '<Offset parameters>'

### Create Nonlinear ARX Model Without Nonlinear Mapping Function

Construct an `idnlarx`

model that uses only linear mapping in the output function. An argument value of `[]`

is equivalent to an argument value of `idLinear`

.

sys = idnlarx([2 2 1],[])

sys = Nonlinear ARX model with 1 output and 1 input Inputs: u1 Outputs: y1 Regressors: Linear regressors in variables y1, u1 Output function: Linear with offset Sample time: 1 seconds Status: Created by direct construction or transformation. Not estimated.

### Create and Combine Regressor Types

Create a regressor set that includes linear, polynomial, periodic, and custom regressors.

Specify `L`

as the set of linear regressors ${\mathit{y}}_{1}\left(\mathit{t}-1\right)$, ${\mathit{u}}_{1}\left(\mathit{t}-2\right)$, and ${\mathit{u}}_{1}\left(\mathit{t}-5\right)$.

L = linearRegressor({'y1','u1'},{1, [2 5]});

Specify `P`

as the set of polynomial regressors ${\mathit{y}}_{2}{\left(\mathit{t}-4\right)}^{2}$, ${\mathit{y}}_{2}{\left(\mathit{t}-5\right)}^{2}$,${\mathit{y}}_{2}{\left(\mathit{t}-6\right)}^{2}$, and ${\mathit{y}}_{2}{\left(\mathit{t}-7\right)}^{2}$.

`P = polynomialRegressor('y2',4:7,2);`

Specify SC as the set of periodic regressors $\mathrm{sin}\left({\mathit{y}}_{1}\left(\mathit{t}-1\right)\right)$, $\mathrm{cos}\left({\mathit{y}}_{1}\left(\mathit{t}-1\right)\right)$, $\mathrm{sin}\left({\mathit{u}}_{1}\left(\mathit{t}-2\right)\right)$, and $\mathrm{cos}\left({\mathit{u}}_{1}\left(\mathit{t}-2\right)\right)$.

SC = periodicRegressor({'y1','u1'},{1,2});

Specify `C`

as the custom regressor $\mathrm{sin}\left({\mathit{y}}_{1}\left(\mathit{t}-1\right){\mathit{u}}_{1}\left(\mathit{t}-2\right)+{\mathit{u}}_{2}\left(\mathit{t}-2\right)\right)$, using the `@`

symbol to create an anonymous function handle.

C = customRegressor({'y1','u1','u2'},{1 2 2},@(x,y,z)sin(x.*y+z));

Combine the regressors into one regressor set `R`

.

R = [L;P;SC;C]

R = [4 1] array of linearRegressor, polynomialRegressor, periodicRegressor, customRegressor objects. ------------------------------------ 1. Linear regressors in variables y1, u1 Variables: {'y1' 'u1'} Lags: {[1] [2 5]} UseAbsolute: [0 0] TimeVariable: 't' ------------------------------------ 2. Order 2 regressors in variables y2 Order: 2 Variables: {'y2'} Lags: {[4 5 6 7]} UseAbsolute: 0 AllowVariableMix: 0 AllowLagMix: 0 TimeVariable: 't' ------------------------------------ 3. Periodic regressors in variables y1, u1 with 1 Fourier terms Variables: {'y1' 'u1'} Lags: {[1] [2]} W: 1 NumTerms: 1 UseSin: 1 UseCos: 1 TimeVariable: 't' UseAbsolute: [0 0] ------------------------------------ 4. Custom regressor: sin(y1(t-1).*u1(t-2)+u2(t-2)) VariablesToRegressorFcn: @(x,y,z)sin(x.*y+z) Variables: {'y1' 'u1' 'u2'} Lags: {[1] [2] [2]} Vectorized: 1 TimeVariable: 't'

Create a nonlinear ARX model.

sys = idnlarx({'y1','y2'},{'u1','u2'},R)

sys = Nonlinear ARX model with 2 outputs and 2 inputs Inputs: u1, u2 Outputs: y1, y2 Regressors: 1. Linear regressors in variables y1, u1 2. Order 2 regressors in variables y2 3. Periodic regressors in variables y1, u1 with W = 1, and 1 Fourier terms 4. Custom regressor: sin(y1(t-1).*u1(t-2)+u2(t-2)) Output functions: Output 1: Wavelet network with number of units chosen automatically Output 2: Wavelet network with number of units chosen automatically Sample time: 1 seconds Status: Created by direct construction or transformation. Not estimated.

### Create Nonlinear ARX Model Using Linear Model

Use a linear ARX model instead of a regressor set to construct a nonlinear ARX model.

Construct a linear ARX model using `idpoly`

.

```
A = [1 -1.2 0.5];
B = [0.8 1];
LinearModel = idpoly(A, B, 'Ts', 0.1);
```

Specify input and output names for the model using dot notation.

LinearModel.OutputName = 'y1'; LinearModel.InputName = 'u1';

Construct a nonlinear ARX model using the linear ARX model.

m1 = idnlarx(LinearModel)

m1 = Nonlinear ARX model with 1 output and 1 input Inputs: u1 Outputs: y1 Regressors: Linear regressors in variables y1, u1 Output function: Wavelet network with number of units chosen automatically Sample time: 0.1 seconds Status: Created by direct construction or transformation. Not estimated.

You can create a linear ARX model from any identified discrete-time linear model.

Estimate a second-order state-space model from estimation data `z1`

.

load iddata1 z1 ssModel = ssest(z1,2,'Ts',0.1);

Construct a nonlinear ARX model from `ssModel`

. The software uses the input and output names that `ssModel`

extracts from `z1`

.

m2 = idnlarx(ssModel)

m2 = Nonlinear ARX model with 1 output and 1 input Inputs: u1 Outputs: y1 Regressors: Linear regressors in variables y1, u1 Output function: Wavelet network with number of units chosen automatically Sample time: 0.1 seconds Status: Created by direct construction or transformation. Not estimated.

### Modify Regressor Assignments to Output Function Components

Modify regressor assignments by modifying the `RegressorUsage`

table.

Construct a nonlinear ARX model that has two inputs and two outputs.

Create the variable names and the regressors.

varnames = {'y1','y2','u1','u2'}; lags = {[1 2 3],[1 2],[1 2],[1 3]}; reg = linearRegressor(varnames,lags);

Create an output function specification `fcn`

that uses `idWaveletNetwork`

for mapping regressors to output `y1`

and `idSigmoidNetwork`

for mapping regressors to output `y2`

. Both mapping objects contain linear and nonlinear components.

fcn = [idWaveletNetwork;idSigmoidNetwork];

Construct the nonlinear ARX model.

output_name = {'y1' 'y2'}; input_name = {'u1' 'u2'}; sys = idnlarx(output_name,input_name,reg,fcn)

sys = Nonlinear ARX model with 2 outputs and 2 inputs Inputs: u1, u2 Outputs: y1, y2 Regressors: Linear regressors in variables y1, y2, u1, u2 Output functions: Output 1: Wavelet network with number of units chosen automatically Output 2: Sigmoid network with 10 units Sample time: 1 seconds Status: Created by direct construction or transformation. Not estimated.

Display the `RegressorUsage`

table.

disp(sys.RegressorUsage)

y1:LinearFcn y1:NonlinearFcn y2:LinearFcn y2:NonlinearFcn ____________ _______________ ____________ _______________ y1(t-1) true true true true y1(t-2) true true true true y1(t-3) true true true true y2(t-1) true true true true y2(t-2) true true true true u1(t-1) true true true true u1(t-2) true true true true u2(t-1) true true true true u2(t-3) true true true true

The rows of the table represent the regressors. The first two columns of the table represent the linear and nonlinear components of the mapping to output `y1`

(`idWaveletNetwork`

). The last two columns represent the two components of the mapping to output `y2`

`(idSigmoidNetwork)`

.

In this table, all the input and output regressors are inputs to all components.

Remove the `y2(t-2)`

regressor from the `y2`

nonlinear component.

sys.RegressorUsage{4,4} = false; disp(sys.RegressorUsage)

y1:LinearFcn y1:NonlinearFcn y2:LinearFcn y2:NonlinearFcn ____________ _______________ ____________ _______________ y1(t-1) true true true true y1(t-2) true true true true y1(t-3) true true true true y2(t-1) true true true false y2(t-2) true true true true u1(t-1) true true true true u1(t-2) true true true true u2(t-1) true true true true u2(t-3) true true true true

The table displays `false`

for this regressor-component pair.

Store the regressor names in `Names`

.

Names = sys.RegressorUsage.Properties.RowNames;

Determine the indices of the rows that contain y`1`

or y`2`

and set the corresponding values of `y1:NonlinearFcn`

to `False`

.

idx = contains(Names,'y1')|contains(Names,'y2'); sys.RegressorUsage{idx,2} = false; disp(sys.RegressorUsage)

y1:LinearFcn y1:NonlinearFcn y2:LinearFcn y2:NonlinearFcn ____________ _______________ ____________ _______________ y1(t-1) true false true true y1(t-2) true false true true y1(t-3) true false true true y2(t-1) true false true false y2(t-2) true false true true u1(t-1) true true true true u1(t-2) true true true true u2(t-1) true true true true u2(t-3) true true true true

The table values reflect the new assignments.

The `RegressorUsage`

table provides complete flexibility for individually controlling regressor assignments.

## More About

### Definition of idnlarx States

The states of an `idnlarx`

object are an ordered list of delayed input
and output variables that define the structure of the model. The toolbox uses this
definition of states for creating the initial state vector that `sim`

, `predict`

, and `compare`

use for simulation and prediction. `idnlarx/linearize`

also uses this definition for linearization of nonlinear ARX
models.

This toolbox provides several options to facilitate how you specify the initial states.
For example, you can use `findstates`

and `data2state`

to search for state values in simulation and prediction
applications. For linearization, use `findop`

. You can also specify the
states manually.

The states of an `idnlarx`

model depend on the maximum delay in each
input and output variable used by the regressors. If a variable *p* has a
maximum delay of *D* samples, then it contributes *D*
elements to the state vector at time *t*:
*p*(*t*–1),
*p*(*t*–2), ...,
*p*(*t*–*D*).

For example, if you have a single-input, single-output `idnlarx`

model.

m = idnlarx([2 3 0],'idWaveletNetwork','CustomRegressors',{'y1(t-10)*u1(t-1)'});

This model has these regressors.

getreg(m)

`ans = `*6x1 cell*
{'y1(t-1)' }
{'y1(t-2)' }
{'u1(t)' }
{'u1(t-1)' }
{'u1(t-2)' }
{'y1(t-10)*u1(t-1)'}

The regressors show that the maximum delay in the output variable `y1`

is 10 samples and the maximum delay in the input `u1`

is two samples. Thus,
this model has a total of 12 states:

```
X(t) =
[y1(t-1),y2(t-2),…,y1(t-10),u1(t-1),u1(t-2)]
``` | (1) |

**Note**

The state vector includes the output variables first, followed by input variables.

As another example, consider the 2-output and 3-input model.

m = idnlarx([2 0 2 2 1 1 0 0; 1 0 1 5 0 1 1 0],[idWaveletNetwork; idLinear]);

This model has these regressors.

getreg(m)

`ans = `*11x1 cell*
{'y1(t-1)'}
{'y1(t-2)'}
{'u1(t-1)'}
{'u1(t-2)'}
{'u2(t)' }
{'u2(t-1)'}
{'u2(t-2)'}
{'u2(t-3)'}
{'u2(t-4)'}
{'u2(t-5)'}
{'u3(t)' }

The maximum delay in output variable `y1`

is two samples. This delay
occurs in the regressor set for output 1. The maximum delays in the three input variables
are 2, 5, and 0, respectively. Thus, the state vector is:

X(t) = [y1(t-1), y1(t-2), u1(t-1), u1(t-2), u2(t-1), u2(t-2), u2(t-3), u2(t-4), u2(t-5)]

Variables `y2`

and `u3`

do not contribute to the state
vector because the maximum delay in these variables is zero.

A simpler way to determine states by inspecting regressors is to use `getDelayInfo`

, which returns the maximum delays in all I/O variables across all
model outputs. For the multi-input multi-output model `m`

,
`getDelayInfo`

returns:

maxDel = getDelayInfo(m)

`maxDel = `*1×5*
2 0 2 5 0

`maxDel`

contains the maximum delays for all input and output variables
in the order (`y1`

, `y2`

, `u1`

,
`u2`

, `u3`

). The total number of model states is
`sum(maxDel) = 9`

.

The set of states for an `idnlarx`

model is not required to be
minimal.

## Version History

**Introduced in R2007a**

### R2023b: New neural network mapping object creates neural networks from both Statistics and Machine Learning Toolbox and Deep Learning Toolbox

The `idNeuralNetwork`

mapping object creates neural networks using both the regression networks of Statistics and Machine Learning Toolbox and the shallow or deep networks of Deep Learning Toolbox. This mapping object replaces and enhances the functionality of `idFeedforwardNetwork`

, which is limited to the shallow networks of Deep Learning Toolbox. For more information, see `idNeuralNetwork`

.

### R2022a: Normalization and regressor selection moved from mapping object properties to `idnlarx`

object

Information related to data normalization was moved from the mapping object properties
to the `idnlarx`

`Normalization`

property. In addition, the regressor-selection process for the
mapping objects was moved to `idnlarx`

. The model now passes the actual
regressor names rather than the selection indices to the mapping object.

### R2021b: Use of previous `idnlarx`

and `idnlhw`

mapping object names is not recommended.

Starting in R2021b, the mapping objects (also known as nonlinearities) used in the nonlinear components of the `idnlarx`

and `idnlhw`

objects have been renamed. The following table lists the name changes.

Pre-R2021b Name | R2021b Name |
---|---|

`wavenet` | `idWaveletNetwork` |

`sigmoidnet` | `idSigmoidNetwork` |

`treepartition` | `idTreePartition` |

`customnet` | `idCustomNetwork` |

`saturation` | `idSaturation` |

`deadzone` | `idDeadZone` |

`pwlinear` | `idPiecewiseLinear` |

`poly1d` | `idPolynomial1D` |

`unitgain` | `idUnitGain` |

`linear` | `idLinear` |

`neuralnet` | `idFeedforwardNetwork` |

Scripts with the old names still run normally, although they will produce a warning. Consider using the new names for continuing compatibility with newly developed features and algorithms. There are no plans to exclude the use of these object names at this time.

### R2021a: Use of previous `idnlarx`

properties is not recommended.

Starting in R2021a, several properties of `idnlarx`

have been modified or
replaced.

These changes affect the syntaxes in both `idnlarx`

and `nlarx`

. The use of the pre-R2021a properties in the following table is
discouraged. However, the software still accepts calling syntaxes that include these
properties. There are no plans to exclude these syntaxes at this time. The command syntax
that uses ARX model orders continues be a recommended syntax.

Pre-R2021a Property | R2021a Property | Usage |
---|---|---|

ARX model orders `na,nb,nk` | `Regressors` , which can include `linearRegressor` , `polynomialRegressor` , and `customRegressor` objects. |
You can no longer change
order values in an existing |

`customRegressors` | `Regressors` | Use `polynomialRegressor` or `customRegressor` to create regressor objects and add the objects to the
`Regressors` array. |

`NonlinearRegressors` | `RegressorUsage` | `RegressorUsage` is a table that contains regressor
assignments to linear and nonlinear output components. Change assignments by
modifying the corresponding `RegressorUsage` table
entries. |

`Nonlinearity` | `OutputFcn` | Change is in name only. Property remains an object or an array or objects that map regressor inputs to an output. |

## See Also

`nlarx`

| `linearRegressor`

| `polynomialRegressor`

| `periodicRegressor`

| `customRegressor`

| `idnlarx/findop`

| `getreg`

| `idnlarx/linearize`

| `pem`

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